"We believe that we're trying to get a good down payment on the wall. It takes a long time to build," Shelby told reporters after his meeting at the White House.

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The meeting came amid the ongoing dispute over Trump’s “zero tolerance” policy at the southern border, which is leading to the separation of immigration families. Trump administration officials have repeatedly said that only Congress can end the practice, and the president has called for border wall funding as a necessary part of any immigration legislation.

Shelby added that senators are expected to include $1.6 billion for border barriers for fiscal 2019 in the DHS appropriations bill, and that he hoped it would be in new construction.

Pressed if Trump would like still more funding, Shelby said that he was "sure [Trump] would like a lot more. All presidents do."

"[But] we're not going to do that right now. We're trying to work to move the bill, but we believe this is a good down payment. This is a lot more than he had last year," Shelby said.

"We had a great meeting. We talked about our shared vision of strong border security, including technology, border agents and a wall," Capito told reporters after the meeting.

Pressed if Trump was OK with $1.6 billion, Capito sidestepped, saying lawmakers "are working on it."

Congress has until Sept. 30 to pass government funding legislation and prevent another shutdown. Trump has warned that he wants Congress to provide funding for the U.S-Mexico border wall and threatened to veto a mammoth omnibus bill in March because it did not provide the full $25 billion in funding.

"I am considering a VETO of the Omnibus Spending Bill based on the fact that the 800,000 plus DACA recipients have been totally abandoned by the Democrats (not even mentioned in Bill) and the BORDER WALL, which is desperately needed for our National Defense, is not fully funded," Trump tweeted at the time.

But full funding for the border wall is unlikely to pass the Senate without a fix for the immigration status of so-called Dreamers, immigrants who came to the country illegally as children.

Shelby added on Monday that Trump did not indicate if he would veto a government funding bill that didn't include more border wall money.

"We had a general conversation about regular order, nobody wanted a government shutdown," Shelby told reporters. "We're working toward that end."